The Mirror reports that Astori was on the shopping list before Dejan Lovren joined the Saints in January.
The 26-year-old has also been strongly linked with the likes of AC Milan and Everton.
The £10million-rated defender played for Italy in the Confederations Cup in Brazil last summer.
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Chelsea have struggled to build momentum in the opening stages of the campaign and are currently looking over their shoulders at the relegation zone.
Who of Europe’s top strikers will move to England this summer?